In their famous book, James and Kerber stated in 1981: "The inner tensor product of two ordinary irreducible representations of a symmetric group is in general reducible, so the question arises how to evaluate its decomposition into irreducible constituents." This is equivalent to a central problem of algebraic combinatorics, namely to decompose the Kronecker product of Schur functions, which are a fundamental basis of the algebra of symmetric functions. The search for an algorithmic description of the Kronecker coefficients was  initiated already about 80 years ago, but positive combinatorial formulae are known so far only for representations to very special partitions. In recent years the Kronecker problem has gained further attention due to connections with quantum information theory and geometric complexity theory.

In the talk, we will focus on classification problems for products with restricted Kronecker coefficients. Tensor products where the decomposition into irreducibles has few components or is multiplicity-free will receive special attention.